def test_frozen(self):
# Test that the frozen and non-frozen inverse Wishart gives the same
# answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim) + 1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim - 1) / 2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim) + (i + 1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim - 1) / 2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
iw = invwishart(df, scale)
assert_equal(iw.var(), invwishart.var(df, scale))
assert_equal(iw.mean(), invwishart.mean(df, scale))
assert_equal(iw.mode(), invwishart.mode(df, scale))
assert_allclose(iw.pdf(x), invwishart.pdf(x, df, scale))
def test_frozen(self):
# Test that the frozen and non-frozen inverse Wishart gives the same
# answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim)+1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim)+(i+1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
iw = invwishart(df, scale)
assert_equal(iw.var(), invwishart.var(df, scale))
assert_equal(iw.mean(), invwishart.mean(df, scale))
assert_equal(iw.mode(), invwishart.mode(df, scale))
assert_allclose(iw.pdf(x), invwishart.pdf(x, df, scale))
def test1():
np.random.seed(0)
nside = 8
ell = np.arange(3. * nside)
Cltrue = np.zeros_like(ell)
Cltrue[2:] = 1 / ell[2:]**2
m = hp.synfast(Cltrue, nside)
Clhat = hp.anafast(m)
sigma_l = (2 * ell + 1) * Clhat
sigma_l = Clhat
Cl = np.linspace(5e-3, 5, 1000)
sl = sigma_l[2]
l = ell[2]
y = np.exp(-(2 * l + 1) * sl / (2 * Cl)) / np.sqrt(Cl**(2 * l + 1))
plt.semilogx(Cl, y / y.max())
alpha = (2 * l - 1) / 2
beta = (2 * l + 1) * sl / 2
pdf = invgamma.pdf(Cl, alpha, scale=beta)
plt.semilogx(Cl, pdf / pdf.max())
pdf = invwishart.pdf(Cl, df=alpha * 2, scale=beta * 2)
plt.semilogx(Cl, pdf / pdf.max())
plt.axvline(Clhat[2])
plt.axvline(Cltrue[2], color='k', linestyle='--')
plt.savefig('test1.pdf')
#plt.show()
plt.close()
return
def pdf(self, x):
"""
PDF for Inverse Wishart prior
Parameters
----------
x : float
Covariance matrix for which the prior is being formed over
Returns
----------
- p(x)
"""
return invwishart.pdf(X, df=self.v, scale=self.Psi)
def pdf(self, X):
"""
PDF for Inverse Wishart prior
Parameters
----------
x : float
Covariance matrix for which the prior is being formed over
Returns
----------
- p(x)
"""
return invwishart.pdf(X, df=self.v, scale=self.Psi)
cov_est = s / (j1 + N)
print(cov_est)
print("Independent Jeffrey's prior")
cov_est = s / (j2 + N)
print(cov_est)
# Q4 Monte carlo estimation
M = [1000, 10000, 100000]
print("Monte Carlo Bayesian Estimation")
for m in M:
print("m =", m)
sigmas = invwishart.rvs(v0, d0, size=m)
lx = invwishart.pdf(sigmas.T, df=N - 3, scale=s)
A = np.dot(sigmas.T, lx) / np.sum(lx)
print(A)
d0 = np.array([[2, 0], [0, 4]])
print(" part b")
for m in M:
print("m =", m)
sigmas = invwishart.rvs(v0, d0, size=m)
lx = invwishart.pdf(sigmas.T, df=N - 3, scale=s)
A = np.dot(sigmas.T, lx) / np.sum(lx)
print(A)
# Q5 Gibbs Sampling
print("Gibbs sampler")
k_ = 500 # number of grid points
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step01): Compute the expectation and standard deviations of Sigma2 and B
# +
exp_sigma2 = invwishart.mean(v_pri, v_pri * sigma2_pri)
std_sigma2 = np.sqrt(invwishart.var(v_pri, v_pri * sigma2_pri))
exp_beta = beta_pri
std_beta = np.sqrt(sigma2_pri / (sigma2_zpri * t_pri) * v_pri / (v_pri - 2.))
# -
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step02): Compute the marginal pdf of Sigma2
s = np.linspace(0.1, exp_sigma2 + 3 * std_sigma2, k_)
f_sigma2 = invwishart.pdf(s, v_pri, v_pri * sigma2_pri)
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step03): Compute the marginal pdf of B
b = np.linspace(exp_beta - 3 * std_beta, exp_beta + 3 * std_beta, k_)
f_beta = t.pdf((b - beta_pri) / np.sqrt(sigma2_pri / (sigma2_zpri * t_pri)),
v_pri) / np.sqrt(sigma2_pri / (sigma2_zpri * t_pri))
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step04): Compute the joint pdf of B and Sigma2
f_joint = np.zeros((k_, k_))
for k in range(k_):
f_joint[:, k] = norm.pdf(b, beta_pri, np.sqrt(
s[k] / (sigma2_zpri * t_pri))) * f_sigma2[k]
# ## Plots
def pdf(self,X):
return invwishart.pdf(X, df=self.v, scale=self.Psi)
import matplotlib.pyplot as plt from scipy.stats import invwishart, invgamma x = np.linspace(0.01, 1, 100) iw = invwishart.pdf(x, df=6, scale=1) iw[:3] # array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03]) ig = invgamma.pdf(x, 6/2., scale=1./2) ig[:3] # array([ 1.20546865e-15, 5.42497807e-06, 4.45813929e-03]) plt.plot(x, iw) # The input quantiles can be any shape of array, as long as the last # axis labels the components.
exp_sigma2_pri = invwishart.mean(v_pri, v_pri * sigma2_pri)
std_sigma2_pri = np.sqrt(invwishart.var(v_pri, v_pri * sigma2_pri))
exp_sigma2_pos = invwishart.mean(v_pos, v_pos * sigma2_pos)
std_sigma2_pos = np.sqrt(invwishart.var(v_pos, v_pos * sigma2_pos))
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_posterior_niw-implementation-step04): Compute marginal pdfs of the sample, prior and posterior distributions of Sigma2
# +
s_max = np.max([
exp_sigma2_hat + 3. * std_sigma2_hat, exp_sigma2_pri + 3. * std_sigma2_pri,
exp_sigma2_pos + 3. * std_sigma2_pos
])
s = np.linspace(0.01, s_max, k_) # grid
f_sigma2_hat = wishart.pdf(s, t_ - 1, sigma2_hat / t_) # sample pdf
f_sigma2_pri = invwishart.pdf(s, v_pri, v_pri * sigma2_pri) # prior pdf
f_sigma2_pos = invwishart.pdf(s, v_pos, v_pos * sigma2_pos) # posterior pdf
# -
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_posterior_niw-implementation-step05): Compute the pdf of the sample, prior and posterior distributions of M
# +
m_min = np.min([
mu_hat - 3. * np.sqrt(sigma2_hat / t_),
mu_pri - 3. * np.sqrt(sigma2_pri / t_pri),
mu_pos - 3. * np.sqrt(sigma2_pos / t_pos)
])
m_max = np.max([
mu_hat + 3. * np.sqrt(sigma2_hat / t_),
mu_pri + 3. * np.sqrt(sigma2_pri / t_pri),
mu_pos + 3. * np.sqrt(sigma2_pos / t_pos)
k_ = 500 # number of grid points
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_prior_niw-implementation-step01): Compute the expectation and standard deviations of Sigma2 and M
# +
exp_sigma2 = invwishart.mean(v_pri, v_pri * sigma2_pri) # expectation
std_sigma2 = np.sqrt(invwishart.var(v_pri, v_pri * sigma2_pri)) # std
exp_m = mu_pri # expectation
std_m = np.sqrt((sigma2_pri / t_pri) * (v_pri / (v_pri - 2.))) # std
# -
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_prior_niw-implementation-step02): Compute the marginal pdf of Sigma2
y = np.linspace(0.1, exp_sigma2 + 3 * std_sigma2, k_) # grid
f_sigma2 = invwishart.pdf(y, v_pri, v_pri * sigma2_pri) # pdf
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_prior_niw-implementation-step03): Compute the marginal pdf of M
x = np.linspace(exp_m - 3 * std_m, exp_m + 3 * std_m, k_) # grid
f_m = t.pdf((x - mu_pri) / np.sqrt(sigma2_pri / t_pri), v_pri) / \
np.sqrt(sigma2_pri / t_pri) # pdf
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_prior_niw-implementation-step04): Compute the joint pdf of M and Sigma2
f_joint = np.zeros((k_, k_))
for k in range(k_):
f_joint[k, :] = norm.pdf(x, mu_pri, np.sqrt(y[k] / t_pri)) * f_sigma2[k]
# ## Plots
std_sigma2_hat = np.sqrt(wishart.var(t_ - 1, sigma2_hat / t_))
exp_sigma2_pri = invwishart.mean(v_pri, v_pri * sigma2_pri)
std_sigma2_pri = np.sqrt(invwishart.var(v_pri, v_pri * sigma2_pri))
exp_sigma2_pos = invwishart.mean(v_pos, v_pos * sigma2_pos)
std_sigma2_pos = np.sqrt(invwishart.var(v_pos, v_pos * sigma2_pos))
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_posterior_niw-implementation-step06): Compute marginal pdfs of the sample, prior and posterior distributions of Sigma2
# +
s_max = np.max([exp_sigma2_hat + 2. * std_sigma2_hat,
exp_sigma2_pri + 2. * std_sigma2_pri,
exp_sigma2_pos + 2. * std_sigma2_pos])
s = np.linspace(0.01, s_max, k_)
f_sigma2_hat = wishart.pdf(s, t_ - 1, sigma2_hat / t_)
f_sigma2_pri = invwishart.pdf(s, v_pri, v_pri * sigma2_pri)
f_sigma2_pos = invwishart.pdf(s, v_pos, v_pos * sigma2_pos)
# -
# ## [Step 7](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_posterior_niw-implementation-step07): Compute the mean and standard deviations of the sample, prior and posterior distributions of B
exp_beta_hat = beta_hat
std_beta_hat = np.sqrt(sigma2_hat / (sigma2_zhat * t_))
exp_beta_pri = beta_pri
std_beta_pri = np.sqrt(sigma2_pri / (sigma2_zpri * t_pri) *
v_pri / (v_pri - 2.))
exp_beta_pos = beta_pos
std_beta_pos = np.sqrt(sigma2_pos / (sigma2_zpos * t_pos) *
v_pos / (v_pos - 2.))
# ## [Step 8](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_posterior_niw-implementation-step08): Compute marginal pdfs of the sample, prior and posterior distributions of B